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# Kuiper's test

In statistics, Kuiper's test is closely related to the more well-known Kolmogorov-Smirnov test (or K-S test as it is often called). As with the K-S test, the quantities D+ and D- are computed which represent the maximum deviation above and below of the two cumulative distributions being compared. The trick with Kuiper's test is to use the quantity D+ + D- as the test statistic. This small change makes Kuiper's test as sensitive in the tails as at the median and also makes it invariant on cyclic transformations of the independent variable. The Anderson-Darling test is another test that provides equal sensitivity at the tails as the median, but it does not provide the cyclic invariance.

This invariance makes Kuiper's test invaluable when testing for variations by time of year or day of the week or time of day. One example would be to test the hypothesis that computers fail more in some parts of the year than others. To test this, we would collect the dates on which the test set of computers had failed and build a cumulative distribution. The null hypothesis is that the failures are uniformly distributed. Kuiper's statistic does not change if we change the beginning of the year and doesn't require that we bin failures into months or anything like that.

A test like this would, however, tend to miss the fact that failures occur only on weekends since weekends are spread throughtout the year. This inability to distinguish distributions with a comb-like shape from continuous distributions is a key problem with all statistics based on a variant of the K-S test.

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